On two recurrence relations involving central Delannoy, small and large Schröder numbers

Abstract We present two recurrence relations involving the central Delannoy numbers D n , the small Schröder numbers s n , and the large Schröder numbers S n using combinatorial arguments in terms of lattice paths. The first relation expresses D n in terms of smaller central Delannoy numbers and large Schröder numbers, and it is not so well known. The second relation expresses s n in terms of smaller small Schröder numbers and large Schröder numbers, and it is a completely new. Using the known recurrence for S n , together with the second relation and the mathematical induction, we provide the new proof of the identity S n = 2 s n , for all natural numbers n .